Anomalous U (1) Gauge Bosons as Light Dark Matter in String Theory

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LMU-ASC 34/20

Anomalous U (1) Gauge Bosons as Light Dark Matter in String Theory

Luis A. Anchordoqui,^{1, 2, 3} Ignatios Antoniadis,^{4, 5} Karim Benakli,⁴ and Dieter L¨ust^{6, 7}

1Department of Physics and Astronomy, Lehman College, City University of New York, NY 10468, USA

2Department of Physics, Graduate Center, City University of New York, NY 10016, USA

3Department of Astrophysics, American Museum of Natural History, NY 10024, USA

4Laboratoire de Physique Théorique et Hautes Énergies - LPTHE Sorbonne Université, CNRS, 4 Place Jussieu, 75005 Paris, France

5Albert Einstein Center, Institute for Theoretical Physics University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland

6Max–Planck–Institut f¨ur Physik, Werner–Heisenberg–Institut, 80805 M¨unchen, Germany

7Arnold Sommerfeld Center for Theoretical Physics Ludwig-Maximilians-Universität München, 80333 München, Germany

Present experiments are sensitive to very weakly coupled extra gauge symmetries which motivates further in- vestigation of their appearance in string theory compactifications and subsequent properties. We consider exten- sions of the standard model based on open strings ending on D-branes, with gauge bosons due to strings attached to stacks of D-branes and chiral matter due to strings stretching between intersecting D-branes. Assuming that the fundamental string mass scale saturates the current LHC limit and that the theory is weakly coupled, we show that (anomalous)U(1) gauge bosons which propagate into the bulk are compelling light dark matter candidates. We comment on the possible relevance of theU(1) gauge bosons, which are universal in intersecting D-brane models, to the observed 3σexcess in XENON1T.

INTRODUCTION

The primary objective of the High Energy Physics (HEP) program is to find and understand what physics may lie beyond the Standard SU(3)C ⊗SU(2)L ⊗U(1)Y Model (SM), as well as its connections to gravity and to the hidden sector of particle dark matter (DM). This objective is pursued in several distinct ways. In this Letter, we explore one possible pathway to join the vertices of the HEP triangle using string compactifications with large extra dimensions [1], where sets of D-branes lead to chiral gauge sectors close to the SM [2, 3].

D-branes provide a nice and simple realization of non- abelian gauge symmetry in string theory. A stack ofNidenti- cal parallel D-branes eventually generates aU(N) theory with the associated U(N) gauge group where the corresponding gauge bosons emerge as excitations of open strings ending on the D-branes. Chiral matter is either due to strings stretching between intersecting D-branes, or to appropriate projections on strings in the same stack. Gravitational interactions are described by closed strings that can propagate in all dimensions; these comprise parallel dimensions extended along the D-branes and transverse ones.

String compactifications could leave characteristic foot- prints at particle colliders:

• the emergence of Regge recurrences at parton collision energies √

s∼string mass scale≡Ms=1/√

α⁰[4–6];

• the presence of one or more additionalU(1) gauge symmetries, beyond theU(1)Yof the SM [7–9].

Herein we argue that the (anomalous) U(1) gauge bosons that do not partake in the hypercharge combination could become compelling dark matter candidates. Indeed, as noted elsewhere [10] these gauge fields could live in the bulk and the four-dimensionalU(1) gauge coupling would become in- finitesimally small in low string scale models,g ∼ M_s/M_Pl, whereM_Pl is the Planck mass (for previous investigations in different regions of parameters and different string scenarios,

see for example [11–13]). Note that for typical energiesEof the order of the electron mass, the value ofg is still bigger that the gravitational coupling∼ E/MPl, and the strength of the new force would be about 10⁷ times stronger than gravity, where we have taken Ms ∼ 8 TeV, saturating the LHC bound [14].

To develop some sense for the orders of magnitude in- volved, we now make contact with the experiment. The XENON1T Collaboration has recently reported a surplus of events in 1 . electronic recoils/keV . 7, peaked around 2.8 keV [15]. The total number of events recorded within this energy window is 285, whereas the expected background is 232±15. Taken at face value this corresponds to a significance of roughly 3σ, but unknown backgrounds from tri- tium decay cannot be reliably ruled out [15]. Although the excess is not statistical significant, it is tempting to imagine that it corresponds to a real signal of new physics. A plethora of models have already been proposed to explain the excess, in which the DM particle could be either the main component of the abundance in the solar neighborhood, nDM ∼ 10⁵(mDM/2.8 keV)⁻¹ cm⁻³, or else a sub-component of the DM population. Absorption of a∼ 2.8 keV mass dark vector boson that saturates the local DM mass density provides a good fit to the excess for aU(1)X gauge coupling to electrons ofgX,eff ∼2×10⁻¹⁶−8×10⁻¹⁶[15–20]. For such small masses and couplings, the cosmological production should be non-thermal [17], avoiding constraints from structure forma- tion [21, 22]. Leaving aside attempts to fit the XENON1T excess, we might consider a wider range of dark photon masses and couplings. For light and very weakly coupled dark photons, the cooling of red giants and horizontal branch stars give stronger or similar bounds on g_X,e_ff than direct detec-

arXiv:2007.11697v2 [hep-th] 9 Oct 2020

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tion experiments [23, 24].¹ For instance, rescaling the bounds quoted in [19] leads to an upper boundgX,eff &10⁻¹⁶−10⁻¹⁴ for mX varying from 10 to 100 keV. As an example, if we takemX ∼15 keV in agreement with the bound of∼ 5 keV [21, 22], the upper bound is aboutgX,eff .5×10⁻¹⁶. Obtaining such small values of masses and couplings for the dark photon are challenging as we will show.

GENERATING THE SMALLU(1)XCOUPLINGS Open string models

We start from ten-dimensional type I string theory compactified on a six-dimensional space of volumeV6M⁶_s. The relation between the Planck mass, the string scale, the string couplingg_s, and the total volume of the bulkV₆M⁶_s reads:

M_Pl² = 8

g²_s M⁸_s V6

(2π)⁶. (1)

A hierarchy between the Planck and string scales can be due to either a large volume V6M⁶_s 1 or a very small string coupling. We discuss these two possibilities successively.

From now on, we denote byd the total number of dimensions that are large. For simplicity, we assume that they have a common radiusR while the other 6−d dimensions have a radiusM⁻¹_s . Obviously, the latter have no more a classical geometry and supergravity description as space dimensions;

they represent new degrees of freedom that have a stringy description, for example through the corresponding world-sheet conformal field theories. The couplings of all light states are under control and our formulae still hold. TheU(1)X gauge fields live on a D(3+δ_X)-brane that wraps aδ_X-cycle of vol- umeVX, while its remaining four dimensions extend into the uncompactified space-time. The corresponding gauge coupling is given by:

g²_X= (2π)^δ^X⁺¹gs

VX M^δs^X

. (2)

Assuming all theδ_X-cycles are sub-spaces of internaldlarge dimensions with the same radius, the substitution of (1) into (2) leads to:

g²_X =2πg_s 8 g²_s

!δX/d

Ms

MPl

!2δX/d

. (3)

It is straightforward to see that to realize the weakest gauge interaction the volume seen byU(1)X must exhaust the total

1A point worth noting at this juncture, however, is that there are several stellar systems that exhibit a mild preference for an over-efficient cooling mechanism when compared to theoretical models [25]. Thus, the argument can be turned around and the anomalous cooling could be interpreted as evidence forU(1)Xproduction in dense star cores.

large internal volume suppressing the strength of gravitational interactionsδX =d(as in [16]), yielding

gX = s

16π gs

M_s MPl

∼4×10⁻¹⁴ 0.2 gs

!1/2 M_s 10 TeV

, (4)

where we have taken as reference values gs = 0.2 and Ms & 10 TeV. The latter is a conservative bound from non- observation of stringy excitations at colliders [14] while a slightly stronger bound of order, but model dependent, can be obtained from limits on dimension-six four-fermion operators [26–29]. As forgswe will consider that it is in the range 0.01−0.2, and could be fixed after a careful study of running of the gauge couplings. In the case of toroidal compactifications, the internal six-dimensional volume is expressed in terms of the parallel and transversal radii as

V6=(2π)⁶

dk

Y

i=1

R^k_i

d⊥

Y

j=i

R^⊥_j , (5) where now for each stack of Dp-branes we identify the corre- spondingdk=δ. For instance, if the SM arise from D3-branes and theU(1)X from D7-branes with an internal space having four large dimensions all parallel to the D7-brane world- volume (δX =d=4), we get forgXthe result in (4).

Little strings models

Another possibility for engineering extremely weak extra gauge symmetries is to consider a scenario which allows very small value ofg_s. Such a possibility is provided by small instantons [30, 31] or Little String Theory (LST) [32, 33] where we localize the SM gauge group on Neuveu-Schwarz (NS) branes (dual to the D-branes).

In the case of LST [32, 33], we start with a compactification on a six-dimensional space of volumeV6with the Planck mass given by (1) (up to a factor 2 in the absence of an orientifold).

The internal space is taken as a product of a two-dimensional space, of volumeV2, times a four-dimensional compact space, of volumeV4. However, instead of D-brane discussed above, we assume that the SM degrees of freedom emerge on a stack of NS5-branes wrapping the two-cycle of volumeV2. We take for simplicity this to be a torus made of two orthogonal circles with radiiR1 andR2. The corresponding (tree-level) gauge coupling is given by:

g²_SM=R1

R2

(Type IIA) and g²_SM= 1

R1R2M²_s (Type IIB);

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M²_Pl= 8

g²_s M⁸_s V2V4

(2π)⁶ , and g²_X = (2π)⁷gs

V₂V₄ M⁶_s . (7)

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Now, taking all the internal space radii to be of the order of the string length,M⁶_sV2V4'(2π)⁶, leads to:

g_X'

√32π rM_s

MPl

∼5×10⁻⁷ M_s

10 TeV ^1/2

. (8)

Note however that theU(1)_Xfrom a D-brane does not interact directly with the electrons of the SM on the NS5-brane. Such interaction could arise via a closed string exchange which is likely to be suppressed by two powers of the string coupling, leading to an effective interaction of the order of 10⁻¹⁴.

Small instanatons models

In heterotic strings compactified on K3, of volume VK3

fibered over a two-dimensional baseP¹ of volumeV_P¹ with integrated volume<VK3VP¹>, the Planck mass reads:

M_Pl² =64π

g²_s M⁸_s <VK3VP¹ > . (9) Taking the limit of instanton small size leads to emergence of a gauge group, identified with the SM one, supported at particular points onK3 . The corresponding gauge coupling reads:

g²_SM= 2π²

M²_s <V_P1>, (10) implying that to give phenomenologically acceptable values, the compactification radius should remain of order of the string scale. TheU(1)X is identified within the bulk theory descending from the ten-dimensional gauge symmetry:

gX= gs

2

1 M³_s√

<VK3V_P1 >=4√ πMs

M_Pl ∼6×10⁻¹⁴ Ms

10 TeV. (11) Taking<V_K3V_P1 >'<V_K3 ><V_P1 >, we see that the weak- ness of gravitational interactions, and a consequence of the U(1)_X coupling, can be due either to a large volume of the K3 internal space or to a small string coupling:<VK3 >^1/4∼ GeV⁻¹orgs∼10⁻¹³forMs∼10 TeV.

DARK PHOTON MASS GENERATION

We turn now to the generation of a mass for the dark photon.

Higgs mechanism

Let’s denote by vX the vacuum expectation value for the HiggshXthat breaks theU(1)Xsymmetry. The simplest quar- tic potential−µ²_Xh²_X+λXh⁴_X leads tovX =µX/√

2λX, a Higgs mass of orderµX and a mass for the dark photon

m_X= g_Xµ_X

√2λ_X = p2πg_s 8 g²_s

!δ/2d

M_s MPl

!δ/d

v_X. (12)

This gives ford=δ=6:

mX ∼ 0.2 gs

!1/6 Ms

1000 TeV ² vX

Ms

!

keV. (13) Taking vX ' Ms, this leads to a mass of order 0.1 to 1.4×10³ eV when varying Ms from 10 to 1000 TeV, and gsfrom 0.2 to 0.02. For this region of the parameter space, the gauge coupling varies in the range 4×10⁻¹⁴ . gX . 2×10⁻¹¹. Higher photon masses are of course easier to obtain with smaller number ofdkdimensions. For example, an M_s∼10 TeV, andM_s∼100 TeV lead respectively tom_X ∼6 keV,g_X ∼6 ×10⁻¹⁰, andm_X ∼270 keV,g_X ∼6 ×10⁻⁹for δX=4,d=6 andgs∼0.2 .

St ¨uckelberg mechanism

Another possibility is that the abelian gauge field U(1)X

becomes massive via a St¨uckelberg mechanism as a consequence of a Green-Schwarz (GS) anomaly cancellation [34, 35], which is achieved through the coupling of twisted Ramond-Ramond axions [36, 37]. The mass of the anomalous²U(1)_Xcan be unambiguously calculated by a direct one- loop string computation. Assuming theU(1)_X arises from a brane wrappingδ_Xdimensions among thedlarge dimensions, it is given by

mX=κ s

V_aM²_s

V_XM^δ_sMs= κ (2π)^δx²⁻²







√ 8 g_s

Ms

MPl







δX−δa d

Ms, (14) whereκis the anomaly coefficient (which is in general an ordinary loop suppressed factor),Vais the two-dimensional internal volume corresponding to the propagation of the axion field [10] andδ_ais the number of large dimensions inVa. For δ_a =0, it leads to:

mX= κ (2π)^δx²⁻²







√ 8 gs

M_s MPl







δ_X/d

Ms, (15) which gives∼0.8κkeV and∼38κkeV and forMs∼10 TeV andMs∼100 TeV, respectively (δX =4,d=6 andgs=0.2).

For this region of the parameter space, the gauge coupling varies in the range 6×10⁻¹⁰ .gX .6×10⁻⁹. The caseδa=2 andδ_X =d=4 leads to:

mX = κ (2π)







√ 8 gs

Ms

MPl







1 2

Ms∼172κ 0.2

gs

!¹₂ Ms

10 TeV ³2

keV. (16) For a concrete example of such case, consider 2 D7-branes intersecting in two common directions; namely, D7₁: 1234

2Note that theU(1) is not necessarily anomalous in four dimensions. A mass can be generated for a non-anomalousU(1) by a six-dimensional (6d) GS term associated to a 6d anomaly cancellation in a sector of the theory.

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andD72: 1256, where 123456 denote the internal six directions. Take now 1234 large and 56 small (order the string scale) compact dimensions. The gauge fields ofD71 have a suppression of their coupling by the 4-dimensional internal volumeVX while the states in the intersection of the two D7 branes see only the 12 large dimensions and give 6 dimensional anomalies, cancelled by an axion living in the same intersection, soVais the volume of 12 only.

U(1)kinetic mixing

We have seen that the tiny couplings are not trivial to obtain and lead often to too small dark photon masses. This issue can be alleviated by resorting to the case where effective smaller couplings ofU(1)X to SM states are obtained when the dark photons do not couple directly to the visible sector, but do it through kinetic mixing with ordinary photons. It can be generated by non-renormalisable operators, but it is natural to assume that it is generated by loops of states carrying charges (q⁽ⁱ⁾,q⁽ⁱ⁾_X) under the twoU(1)’s and having massesm_i:

γX = egX

16π² X

i

q⁽ⁱ⁾q⁽ⁱ⁾_X lnm²_i µ² ≡ egX

16π²CLog (17) whereµ² denotes the renormalization scale³, where we ab- sorbed also the constant contribution. The effective coupling to SM is then:

gX,eff =e_γX= α_emg_X

4π CLog∼6×10⁻⁴gXCLog. (18) We can try to fit both desired values ofgX,eff andmX. For a mass of the dark photon arising from a Higgs mechanism, we determinegx∼mX/Ms, withvX ∼Ms, this constrains:

C_Log ' 1.7×10³g_X,e_ff M_s mX

' 0.05 gX,eff

8×10⁻¹⁶

Ms

100 TeV

mX

2.8 keV −1

.(19) A cancellation in the logarithm can be total, and the contribution appears at higher loops [38], or partial, for instance between particles with (order one) charges (q⁽ⁱ⁾,q⁽ⁱ⁾_X) and (q⁽^j),q⁽_X^j) = −q⁽ⁱ⁾_X) and masses mj = mi + ∆mi j. For

∆m_{i j} m_i, we have an approximation:

CLog∼X

i,j

∆mi j

mi

. (20)

TOWARDS EXPLICIT MODELS

We shall now discuss more explicitly the emergence of such extra abelian gauge groups inD-brane models. The minimal

3In string theory, it is replaced by the string scaleM_s.

embedding of the SM particle spectrum requires at least three brane stacks [39] leading to three distinct models of the type U(3)⊗U(2)⊗U(1) that were classified in [39, 40]. Only one of them, model(C)of [40], has baryon number as a gauge symmetry that guarantees proton stability (in perturbation theory), and can be used in the framework of low mass scale string compactifications. In addition, because the charge associated to theU(1) of U(2) does not participate in the hypercharge combination, U(2) can be replaced by the symplecticSp(1) representation of Weinberg-SalamSU(2)_L, leading to a model with one extraU(1) added to the hypercharge [41]. Note that the abelian factor associated to theU(2) stack of D-branes couples to the lepton doublet, and consequently this anoma- lousU(1) cannot be a good dark matter candidate, because the left-handed neutrinos make it unstable. One can add to these three stacks another D9-brane which will provide theU(1)X

which will mix with the photon through loops of states living in the intersections of the D9 and theU(3) andU(1) stacks.

The darkU(1)X is of course unstable as it decays to three ordinary photons. However, the partial decay width is found to be [42]

ΓX→3γ∼10⁻²⁸ mX

2.8 keV

9 gX,eff

5×10⁻¹⁶ 2

Gyr⁻¹, (21) and so for the range of small gauge coupling considered here, the life-time is big enough to allow it be a viable candidate for dark matter.

Actually, the SM embedding in four D-brane stacks leads to many more models that have been classified in [43, 44]. The total gauge group of interest here,

G = U(3)_C⊗U(2)_L⊗U(1)₁⊗U(1)_X (22)

= SU(3)C⊗U(1)C⊗SU(2)L⊗U(1)L⊗U(1)1⊗U(1)X, contains four abelian factors. The non-abelian structure deter- mines the assignments of the SM particles. The quark doublet Qcorresponds to an open string with one end on the color stack of D-branes and the other on the weak stack. The anti- quarksu^candd^cmust have one of their ends attached to the color branes. The lepton doublet and possible Higgs doublets must have one end on the weak set of branes. Per contra, the abelian structure is not fixed because theU(1)Yboson, which gauges the usual electroweak hypercharge symmetry, could be a linear combination of all four abelian factors. However, herein we restrict ourselves to models in which the bulkU(1)_X does not contribute to the hypercharge, in order to avoid an un- realistically small gauge coupling. Of particular interest here are models(3)and(5)of reference [43]. The general properties of their chiral spectra are summarized in Table I and II.

One can check by inspection that for both models the hypercharge,

q_Y=−1 3q_C+1

2q_L+q₁ for model(3) qY= 2

3qC+1

2qL+q1 for model(5) (23)

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TABLE I: Chiral fermion spectrum of the D-brane model(3).

Fields Representation qC qL q1 qX qY

Q (3,2) 1 1 0 0 ¹₆

u^c (3,¯ 1) −1 0 −1 0 −²₃

d^c (3,¯ 1) −1 0 0 −1 ¹₃

L (1,2) 0 1 −1 0 −¹

2

e^c (1,1) 0 0 1 1 1

TABLE II: Chiral fermion spectrum of the D-brane model(5).

Fields Representation qC qL q1 qX qY

Q (3,2) 1 −1 0 0 ¹₆

u^c (3,¯ 1) −1 0 0 1 −²

3

d^c (3,¯ 1) −1 0 1 0 ¹₃

L (1,2) 0 1 −1 0 −¹

2

e^c (1,1) 0 0 1 1 1

is anomaly free. In addition, theU(1)X is long-lived (because it only couples to thee^cand to eitheru^cord^c) and therefore a viable DM candidate.

SUMMARY

We have investigated the possibility of identification of the light dark photon with one of the ubiquitousU(1) gauge bosons of D-brane string theory constructions. We have first investigated how small the gauge coupling can be made. We found that open strings allow values as weak asgX ∼ O(10⁻¹⁴) for a string scale of orderMs∼ O(10) TeV. For the case of six and four large extra dimensions, the Kaluza-Klein excitations appear above the GeV and MeV scales, respectively. They are very weakly coupled and decay quickly, but one could hope to observe their cumulative effect at TeV scales energies in future collider searches. The case of small instantons leads to similar conclusions with an interesting additional possibility:

no large extra dimension below the TeV but a tiny string coupling. This possibility is also realized in Little String Theory but with a stronger gauge couplinggX ∼ O(10⁻⁷). We have then looked at possible realization of models with dark photon masses in the range of keV. We have found two possibilities.

A stringy St¨uckelberg mechanism generates masses in the de- sirable range in models with four large extra dimensions. If the dark photon mass is generated instead by a (low energy field theoretical) Higgs mechanism, then a kinetic mixing between the visible and darkU(1) allows to get simultaneously the desired mass and coupling strength.

Here, we have left aside a couple of issues whose investigations are beyond the scope of this work. String models typically exhibit a large number of moduli fields that need to be fixed. In particular, the size of extra dimensions and couplings discussed here are vacuum expectation of such fields and all the hierarchies should be dynamically generated (see

[45] for example). Also, obtaining the correct relic density for such weakly coupled particles to form the main component of dark matter is challenging. The dark photons can not be thermally produced and one needs to resort to different mech- anisms. A promising possibility is the proposal of [46], the desired abundance of dark photons can be generated by the quantum fluctuations during inflation for an appropriate set of parameters: the scale of inflation, reheating temperature and coupling to the curvature scalar.

Finally, we have discussed possible D-brane model can accommodate the excess of events with 3σsignificance over background recently observed at XENON1T. The model is fully predictive, and can be confronted with future data from dark matter direct-detection experiments, LHC Run 3 searches, and astrophysical observations.

The work of L.A.A. is supported by the U.S. National Science Foundation (NSF Grant PHY-1620661) and the Na- tional Aeronautics and Space Administration (NASA Grant 80NSSC18K0464). The research of I.A. is funded in part by the “Institute Lagrange de Paris”, and in part by a CNRS PICS grant. The work of K.B. is supported by the Agence Na- tionale de Recherche under grant ANR-15-CE31-0002 “Hig- gsAutomator”. The work of D.L. is supported by the Origins Excellence Cluster. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF or NASA.

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